Injective function: Difference between revisions

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{{Functions}}

In [[mathematics]], an '''injective function''' (also known as '''injection''', or '''one-to-one function'''<ref>Sometimes ''one-one function'', in Indian mathematical education. {{Cite web |title=Chapter 1:Relations and functions |url=https://ncert.nic.in/ncerts/l/lemh101.pdf |via=NCERT |url-status=live |archive-url=https://web.archive.org/web/20231226194119/https://ncert.nic.in/ncerts/l/lemh101.pdf |archive-date= Dec 26, 2023 }}</ref> ) is a [[function (mathematics)|function]] {{math|''f''}} that maps [[Distinct (mathematics)|distinct]] elements of its ___domain to distinct elements of its codomain; that is, {{math|1=''x''₁ ≠ ''x''₂}} implies {{math|''f''(''x''₁) {{≠}} ''f''(''x''₂)}} (equivalently by [[contraposition]], {{math|''f''(''x''₁) {{=}} ''f''(''x''₂)}} implies {{math|1=''x''₁ = ''x''₂}}). In other words, every element of the function's [[codomain]] is the [[Image (mathematics)|image]] of {{em|at most}} one element of its [[Domain of a function|___domain]].<ref name=":0">{{Cite web|url=https://www.mathsisfun.com/sets/injective-surjective-bijective.html|title=Injective, Surjective and Bijective|website=Math is Fun |access-date=2019-12-07}}</ref> The term {{em|one-to-one function}} must not be confused with {{em|one-to-one correspondence}} that refers to [[bijective function]]s, which are functions such that each element in the codomain is an image of ''exactly one'' element in the ___domain.

A [[homomorphism]] between [[algebraic structure]]s is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for [[vector space]]s, an {{em|injective homomorphism}} is also called a {{em|[[monomorphism]]}}. However, in the more general context of [[category theory]], the definition of a monomorphism differs from that of an injective homomorphism.<ref>{{Cite web|url=https://stacks.math.columbia.edu/tag/00V5|title=Section 7.3 (00V5): Injective and surjective maps of presheaves |website=The Stacks project |access-date=2019-12-07}}</ref> This is thus a theorem that they are equivalent for algebraic structures; see {{slink|Homomorphism|Monomorphism}} for more details.

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== Definition ==

{{Dark mode invert|[[file:Injection.svg|thumb|An injective function, which is not also [[Surjective function|surjective]].]]}}

{{Further|topic=notation|Function (mathematics)#Notation}}

Let <math>f</math> be a function whose ___domain is a set <math>X.</math> The function <math>f</math> is said to be '''injective''' provided that for all <math>a</math> and <math>b</math> in <math>X,</math> if <math>f(a) = f(b),</math> then <math>a = b</math>; that is, <math>f(a) = f(b)</math> implies <math>a=b.</math> Equivalently, if <math>a \neq b,</math> then <math>f(a) \neq f(b)</math> in the [[Contraposition|contrapositive]] statement.

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In this case, <math>g</math> is called a [[Retract (category theory)|retraction]] of <math>f.</math> Conversely, <math>f</math> is called a [[Retract (category theory)|section]] of <math>g.</math>

For example: <math>f:\R\rightarrow\R^2,x\mapsto(1,m)^\intercal x</math> is retracted by <math>g:y\mapsto\frac{(1,m)}{1+m^2}y</math>.

Conversely, every injection <math>f</math> with a non-empty ___domain has a left inverse <math>g</math>. It can be defined by choosing an element <math>a</math> in the ___domain of <math>f</math> and setting <math>g(y)</math> to the unique element of the pre-image <math>f^{-1}[y]</math> (if it is non-empty) or to <math>a</math> (otherwise).{{refn|Unlike the corresponding statement that every surjective function has a right inverse, this does not require the [[axiom of choice]], as the existence of <math>a</math> is implied by the non-emptiness of the ___domain. However, this statement may fail in less conventional mathematics such as [[constructive mathematics]]. In constructive mathematics, the inclusion <math>\{ 0, 1 \} \to \R</math> of the two-element set in the reals cannot have a left inverse, as it would violate [[Indecomposability (constructive mathematics)|indecomposability]], by giving a [[Retract (category theory)|retraction]] of the real line to the set {0,1}.}}

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== Other properties ==

{{See also|List of set identities and relations#Functions and sets}}

{{Dark mode invert|[[Image:Injective composition2.svg|thumb|300px|The composition of two injective functions is injective.]]}}

* If <math>f</math> and <math>g</math> are both injective then <math>f \circ g</math> is injective.

* If <math>g \circ f</math> is injective, then <math>f</math> is injective (but <math>g</math> need not be).